Currently the only way to speed up the curse is to get caught by the pink mist roaming the map.
There is plan to make potion affecting time in the near future.
For the secret, there is a total of 4, one for each curse you encounter, and reaching an ending.
You can tell them to the bat... if you didn't annoyed them of course :)
This can be used to check if you got the same curse as previous run, to avoid you losing time if you're going for a different one.